Universal deformations of a smooth marked curve can be constructed for
example using Teichmüller theory [1]} or by Hilbert
scheme methods [2]}. Later we will need an explicit
gluing construction of a universal deformation of a stable marked
curve. This construction seems to be well-known, but the only proof
we could find in the literature is Siebert [3]}. The idea
is to remove small neighborhoods of the nodes, and glue the remaining
components together. A local coordinate near a smooth point \(z\in {\Sigma }\)  is a neighborhood \(U\)  of \(z\)  and a holomorphic
isomorphism \(\kappa \)  of \(U\)  with a neighborhood of 0 in the tangent
line \(T_{z} {\Sigma }\) , whose differential \(T_0 U\rightarrow T_z \Sigma \)  is
the identity.
